So earlier this evening Commander Chris Hadfield, a Canadian astronaut on the ISS, tweeted the question
.@erethizon attention physicists – we’ve been traveling 8 km/sec for 5 months. How much has our time relativity differed from everyone else?
— Chris Hadfield (@Cmdr_Hadfield) May 3, 2013
I gave a quick reply which ended up generating quite a bit of discussion, so I thought I might write up a slightly more detailed answer to the question.
@evan_hadfield gamma = 1/sqrt(1 – v^2/c^2) (v = 8km/s). t = t_0/gamma (t_0 = 5 months). dt = t_0 – (t_0/gamma) = 0.0046s
— timl (@timl) May 3, 2013
The problem as posed relates to the special theory of relativity. Objects in relative motion will measure time running at a different rate. In this case, the astronauts travelling on the ISS are travelling quite quickly relative to an observer on the ground.
We can measure this time dilation using the Lorentz transformation. We begin by taking the speed of the space station, and use it to calculate a quantity called .
We use the values
So this is our time dilation factor, which tells us how much slower the clocks on the space station are running.
Now, if the astronauts have been up there for what we on Earth would measure as 5 months, how much time have the astronauts experienced with their slow running clocks? Well, we can work that out with the equation
Here we let be 5 months, or . So the time experienced by the astronauts will be
This is very close to exactly 5 months, but it’s just a tiny bit smaller. How much smaller is it? Well,
This is much less than a single second! In fact, it’s a touch over 4.6 milliseconds.
So there we go, the astronauts will be 4.6 milliseconds younger than they would be if they’d stayed planted on Earth. The time difference is due to time dilation, a direct consequence of their movement relative to Earth and the theory of special relativity.
But this is only part of the picture! My answer to the original question only took into account the relative speeds of the space station and an observer on Earth. If we delve deeper, we find that gravity also has an effect on the rate at which clocks tick. When standing at sea level, we’re around 6371km from the centre of the Earth. But the space station orbits at around 370km above sea level, which puts them 6741km from the centre of the Earth. Because of this difference in distance, the space station experiences a weaker gravity field than we do here on terra-firma.
The effect of gravity on time can be calculated by using the equations from general relativity. Gravity causes time to run more slowly, so we need to work out how strong this effect is both on Earth, and on the space station. We’ll use the following equation to work out the time dilation factor due to gravity.
In this equation is the Schwarzschild radius of the Earth, which is based on its gravity. It essentially tells us how far away the event horizon would be if the Earth collapsed into a black hole. It turns out to be quite small, just 9mm, or 0.009 metres.
The variable is the distance from the centre of the Earth. Based on the numbers above we can work out that for someone standing at sea level we get
For a person on the space station, we get
We can now work out how long 5 Earth months would take on the space station, by considering the relative clock speeds on Earth and the space station.
So we see that the effect of general relativity causes the astronauts to experience more time than if they’d stayed on Earth! How much more time?
or around 0.5 milliseconds.
So if we put together the effects of both general relatively (gravity) and special relativity (speed differences) we get the total time difference,
So there we have it, the answer to the Commander’s question is 4.1 milliseconds! 5 months in space to travel into the future by less than the blink of an eye. Worth it? Totally!
A heraldic Spring dragon of ice roars rampant off the coast of Newfoundland. twitter.com/Cmdr_Hadfield/…
— Chris Hadfield (@Cmdr_Hadfield) May 2, 2013